The Hilbert space of one link is thus equivalently given by \({\mathcal {H}}_l={\mathcal {F}}_2\times {\mathcal {F}}_2//\mathrm {U}(1)\) . Readers can find more details for the Bargmann space reconstruction of the Hilbert space in [1]}. By taking the closure constraint for nodes into consideration, we conclude that the Hilbert space spanned by the coherent or scaleless spin network states can be represented as
\(L^{\rm hol}_2({\mathbb {C}}^2,\mathrm {d}\mu )^{2|L|}//(\mathrm {SU}(2)^{|N|}\times \mathrm {U}(1)^{|L|})\) .
